Is the sequence $0, 0, 0, 0 …$ geometric? If so how would you define it? In order to define a geometric sequence you need the first term, and the ratio of terms. In this case you could have:
$a = 0$
$r = k$ for some $k \in \mathbb{R}$
Is this still geometric, even though a single unique definition doesn't exist (a non variable $r$)?
EDIT: This is an interesting debate. But you could also say that $0, 0, 0, 0 …$ is an arithmetic sequence. So to all who are saying that it is geometric, can a sequence be both geometric and arithmetic?
Best Answer
Ultimately it will depend on what you consider the definition of a geometric series, as you can see by the other answers here. If the definition relies on calculating a common ratio $r = a_{k+1}/a_k$, then you will run into division by $0$.
You can avoid that problem by saying that a geometric sequence is one whose terms obey the property $a_ka_{k+2} = a_{k+1}^2$. Another way is to define the sequence as $a_k = cr^k$.
Personally, I would say that the sequence $0,0,0,\ldots$ is a geometric sequence in the same way that a point is a circle of radius $0$.